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Math Help - Convergence test for a series with e

  1. #1
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    Convergence test for a series with e

    Hello,

    I have to find out if this series converges:

    \sum_{n=1}^{\infty}{\frac{1}{n}\left(e^\frac{1}{n}-1\right)}

    Any help would be appreciated.
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  2. #2
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    e^x=1+x+O(x^2)

    So \frac{1}{n}\left(e^{\frac 1n}-1\right)=\frac{1}{n}\left(1+\frac{1}{n}+O\left(\fr  ac{1}{n^2}\right)-1\right)=\frac{1}{n^2}+O\left(\frac{1}{n^2}\right).

    Now \sum\left(\frac{1}{n^2}+O\left(\frac{1}{n^2}\right  )\right)<\infty, from which you can conclude that the original series also converged.
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  3. #3
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    That solved my problem. Thank you very much for your help!
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